Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Semi-Log Analysis
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Drawdown Semi-Log Analysis: the MDH Plot (Miller-Dyes-Hutchinson)
In drawdown analysis, the log approximation to the Exponential Integral gives:
which can be written as:
On the MDH plot, one can solve for m and b by reading the coordinates of two
points:
t = 0, pDd = pi, and
t = 1 hr, pDd = p1hr.
( )
(

+
|
.
|

\
|
+ S
r C
k
t
kh
q
p
w t
Dd 86859 . 0 2275 . 3

log log
6 . 162

( ) b t m pDd + = log
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Drawdown Pressure Profile: the MDH Plot
Because the pressure change is proportional to the logarithm of elapsed time when
IARF is reached, a graph of P vs Log t will yield a straight line of slope m.
The effects of wellbore storage and skin are superimposed onto the ideal response as
shown below.
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Drawdown Semi-Log Analysis (contd)
The solution is then:
and
(

+
|
.
|

\
|

= 2275 . 3

log 1513 . 1
1
w t
hr i
r C
k
m
p p
S

m
q
kh
6 . 162
=
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Build-up Analysis
In practice, it is not often possible to conduct drawdown analysis. This is because
drawdown analysis applies to a constant flow rate, a condition which is difficult to
maintain during well tests.
To remedy this shortcoming, it is more practical to analyze build-up periods by
resorting to the the principle of superposition of states.
Modern well testing now offers multiple possibilities to analyze drawdown (flow)
periods by measuring the flow rates downhole during testing. For the interpretation,
the principle of superposition is generalized into a technique called the pressure-flow
convolution.
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
The Principle of Superposition of States
Because of the linearity of the pressure response equation, the response during a buid-
up period is equal to the sum of the responses of two drawdown periods:
- Flow rate q from time t = 0, and
- Flow rate -q from time t = tp (drawdown production time).
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Single Flow Period Superposition for Build-up Analysis
Considering a single flow period of duration tp:
( ) ( ) t t p t p p p p p Dd Dd wf i Bu + + =
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Build-up Semi-Log Analysis: the Horner Plot
For a single flow period, the superposition
function is the Horner time:
On a semi-log plot, the extrapolated pressure
is the static reservoir pressure, provided that
- The reservoir has not entered
depletion regime during the drawdown.
- No late-time effects will affect the
buildup after the end of the buildup (this
is impossible to ascertain without
testing longer).
t
t tp

+
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Build-up Semi-Log Analysis: the Horner Plot (contd)
On the Horner plot, the solution is again:
and
m
q
kh
6 . 162
=
(

+
|
.
|

\
|

= 2275 . 3
r C
k
log
m
p p
1513 . 1 S
w t
wf hr 1
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Generalized Superposition for Build-up Analysis
When the well has been submitted to a series of flow periods prior to build-up, one
must consider a generalized superposition function as follows:
( )
( )
( )
( )
( )

=
=
=

t N i
1 i
i
t N
1 i i
t t ln
q
q q
t Sn
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Multi-Rate Build-up Analysis
When the pressures are plotted versus
Sn(t), the solution is identical to the
case of a single flow period (Horner
plot).
On a semi-log plot, the extrapolated
pressure is the static reservoir pressure,
with the same restrictions as apply to
the Horner plot.
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Pressure Derivative
Log-Log Analysis
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
The Pressure Derivative
Modern well testing advances (1983) have culminated with the introduction of the
Pressure Derivative PD as an indispensable complement to plotting pressures versus
time. By definition:
The Pressure Derivative is the slope of the semi-log plot as shown below.
( ) t d
p d
t
t dLn
p d
' p

=
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Dimensionless Variables
In order to solve the diffusivity equation in typical situations applicable to all possible
values of the physical parameters, one uses dimensionless variables defined as
follows:
Dimensionless distance: in which rw is the wellbore radius.
Dimensionless pressure: in which pi is the initial
pressure.
Dimensionless time: in which t is the elapsed time.
w
D
r
r
r =
) ( 2 p p
q
kh
p i D =

t
r C
k
t
w t
D =
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Homogeneous Reservoir with Wellbore Storage and Skin
Because the skin just adds to the pressure drop in the wellbore, the dimensionless skin
S just adds to the PD function in the solution of the diffusivity equation for IARF:
In physical terms:
( ) | | S t Ln p D D 2 80907 . 0
2
1
+ + =
( )
(

+
|
.
|

\
|
+ S
r C
k
t
kh
q
p t p
w t
i 86859 . 0 2275 . 3

log ) log(
6 . 162

Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
The IARF solution for a well with wellbore storage and skin has been expressed as:
In log-log analysis, it is preferrable to re-write the pressure response as:
in which CD is the dimensionless wellbore storage constant:
( ) | | S t Ln p D D 2 80907 . 0
2
1
+ + =
( ) | |
e
LnC
C
t
Ln p
S
D
D
D
D
2
80907 . 0
2
1
+ + =
h r C
C
C
w t
D
2
=
Homogeneous Reservoir with Wellbore Storage and Skin (contd)
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Type Curves
By plotting the theoretical pressure
response PD versus tD/CD, (instead of
vs tD), one obtains a way of
characterising in a unique way the
IARF solution (for a well with
wellbore storage and skin for
example).
One thus defines an array of type
curves, each curve corresponding to a
value of the sensitivity parameter
CDe**2S.
The inclusion of the pressure derivative
on this plot was a major breakthrough
in well test interpretation.
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Attributes of the Log-Log Plot: Early Time Behaviour
At early times, the pressure response is dominated by the wellbore effect. The
solution of the diffusivity equation is:
This plots as a unit slope on a graph of pD vs tD/CD.
Then
and the derivative matches the pressure response on a unit slope.
This particularity of early time behaviour is one of the most conspicuous features of a
log-log plot in well test interpretation.
D
D
D
C
t
p =
( )
D
D
D
D
D
D
D
D
D
D p
C
t
dt
dp
t
C
t
dLn
dp
p = = = = '
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Attributes of the Log-Log Plot: IARF
The solution of the diffusivity equation for IARF is:
Then
When IARF is reached, the pressure derivative levels off to a plateau on the log-log
plot. The corresponding value of PD is 0.5. Again, this characteristic leveling off of
PD upon reaching IARF is one of the most conspicuous features of the log-log plot in
well test interpretation.
( ) | | S t Ln p D D 2 80907 . 0
2
1
+ + =
( ) 2
1
' = =
D
D
D
t dLn
dp
p
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Type Curve Matching
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Data Set and Type Curve Array
The data collected during a well test are in the form of couples (pressure-time). These
are initially presented as a log-log plot of pressure variations vs elapsed time, with the
computation of the pressure derivative.
Type-curve matching has for objective the superposition of the data set over the array
of type curves corresponding to the model chosen, and the extraction of the test target
parameters.
This will be done by
- shifting the data horizontally (time match).
- shifting the data vertically (pressure match).
- finding the matching type curve (and its derivative) with its characteristic CDe**2S.
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Data Set and Array of Type-Curves
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Matched Data Set
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Pressure Match: Extracting kh
From the expression of dimensionless pressure
one defines the pressure match Mp
Mp is read as the value of pD matching a specific value of p. Then
p
q
kh
pD =
2 . 141
q
kh
p
p
M
D
p
2 . 141
=

=
p M q kh 2 . 141 =
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Time Match: Extracting C
From the expressions of dimensionless time and wellbore storage constant:
one defines the time match Mt
Mt is read as the value of tD/CD matching a specific value of t. Then
t M
kh
C
000295 . 0
=
t
C
kh
C
t
D
D
=

000295 . 0
( )
C
kh
t
C
t
M
D
D
t
000295 . 0
=

=
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Skin Match: Extracting S
One reads the value of Ms on the matching type curve:
Then
with CD calculated from its dimensionless expression:
e
C M
S
D S
2
=
( )
D
S
C
M
Ln S
2
1
=
h r C
C
C
w t
D

8936 . 0

=
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Type-Curve Match Example: Data Set
TCMATCH.WTD (Field Data)
1
10
100
1000
10000
0.001 0.01 0.1 1 10 100 1000
P
r
e
s
s
u
r
e

c
h
a
n
g
e
,

p
s
i
Equivalent time, hrs
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Type-Curve Match Example: Unmatched Overlay
TCMATCH.WTD (Drawdown type curve, Radial equivalent time)
Radial flow, Single porosity, Infinite-acting: Varying CDe2s
0.001
0.01
0.1
1
10
100
0.001 0.01 0.1 1 10 100 1000 10000 100000
D
i
m
e
n
s
i
o
n
l
e
s
s

p
r
e
s
s
u
r
e
Dimensionless time
0.001 0.01 0.1 1 10 100 1000
1
10
100
1000
Equivalent time, hr
P
r
e
s
s
u
r
e

c
h
a
n
g
e
,

p
s
i
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Type-Curve Match Example: Matched in Pressures
TCMATCH.WTD (Drawdown type curve, Radial equivalent time)
Radial flow, Single porosity, Infinite-acting: Varying CDe2s
0.001
0.01
0.1
1
10
100
0.001 0.01 0.1 1 10 100 1000 10000 100000
D
i
m
e
n
s
i
o
n
l
e
s
s

p
r
e
s
s
u
r
e
Dimensionless time
0.001 0.01 0.1 1 10 100 1000
1
10
100
1000
Equivalent time, hr
P
r
e
s
s
u
r
e

c
h
a
n
g
e
,

p
s
i
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Type-Curve Match Example: Matched in Both Times and Pressures
TCMATCH.WTD (Drawdown type curve, Radial equivalent time)
Radial flow, Single porosity, Infinite-acting: Varying CDe2s
0.001
0.01
0.1
1
10
100
0.001 0.01 0.1 1 10 100 1000 10000 100000
D
i
m
e
n
s
i
o
n
l
e
s
s

p
r
e
s
s
u
r
e
Dimensionless time
0.001 0.01 0.1 1 10 100 1000
1
10
100
1000
Equivalent time, hr
P
r
e
s
s
u
r
e

c
h
a
n
g
e
,

p
s
i
Well Test Interpretation Methodology

Gamma Experts
Petroleum Engineering
Yves Chauvel
PRACTICAL RESERVOIR MONITORING
September 2002
Type-Curve Match Example: Extraction of Time, Pressure and Skin Match
TCMATCH.WTD (Drawdown type curve, Radial equivalent time)
Radial flow, Single porosity, Infinite-acting: Varying CDe2s
0.001
0.01
0.1
1
10
100
0.001 0.01 0.1 1 10 100 1000 10000 100000
D
i
m
e
n
s
i
o
n
l
e
s
s

p
r
e
s
s
u
r
e
Dimensionless time
0.001 0.01 0.1 1 10 100 1000
1
10
100
1000
Equivalent time, hr
P
r
e
s
s
u
r
e

c
h
a
n
g
e
,

p
s
i
t
D
/C
D
=1
t
eq
=0.0546 hr
p=262 psi p
D
=10
C
D
e
2s
=7x10
9